L(s) = 1 | − 11.9·5-s − 29.7i·7-s + 41.2i·11-s + 17.8·25-s − 218.·29-s − 257. i·31-s + 355. i·35-s − 540.·49-s − 247.·53-s − 493. i·55-s + 717. i·59-s − 1.20e3·73-s + 1.22e3·77-s + 1.37e3i·79-s − 1.50e3i·83-s + ⋯ |
L(s) = 1 | − 1.06·5-s − 1.60i·7-s + 1.13i·11-s + 0.142·25-s − 1.39·29-s − 1.49i·31-s + 1.71i·35-s − 1.57·49-s − 0.641·53-s − 1.20i·55-s + 1.58i·59-s − 1.93·73-s + 1.81·77-s + 1.95i·79-s − 1.98i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8674975506\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8674975506\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 11.9T + 125T^{2} \) |
| 7 | \( 1 + 29.7iT - 343T^{2} \) |
| 11 | \( 1 - 41.2iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 - 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 218.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 257. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 - 6.89e4T^{2} \) |
| 43 | \( 1 + 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 + 247.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 717. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.20e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.37e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 1.50e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.86e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.105625490838983248839565961587, −7.937173714141226219138724474927, −7.46099794053953213699899028363, −7.04547435704609090066637434050, −5.87668420299500770561412238585, −4.54448033168416620010369596882, −4.17807993642523085578042101748, −3.36731455993187816006000070711, −1.91124110912727304608270272615, −0.67501111275389604850478913148,
0.28145028907757604058026380581, 1.80250136947160897355927547856, 3.00608745918441258100371262923, 3.61649119729244734100142165101, 4.83512576174774037167949190550, 5.63390516142589673387268313456, 6.33214397130152473757046694684, 7.42504779927280494105641843815, 8.210108076920627620301709147030, 8.768377524876712925593332547833